Classical Virasoro irregular conformal block II

We present a new systematic way to evaluate the classical limit of the Virasoro irregular conformal block for arbitrary rank n based on the irregular partition function. In addition, we prove that the classical irregular conformal block has the exponential form as suggested by A. Zamolodchikov and Al. Zamolodchikov for the regular case. We provide an explicit calculation for the rank 2 case in detail.


Introduction
In our recent work [1] which will be called I in the following, we studied the classical limit (NS limit) [2] of Virasoro irregular conformal block (ICB) using the irregular matrix model (IMM) [3] . IMM is a β-deformed one matrix model with a logarithmic as well as a finite number of inverse power potentials. The finite number n of the inverse powers is called the rank of irregular model. It is demonstrated that the classical ICB of rank n can be obtained by using the generalized Mathieu equation. This equation is equivalent to the loop equation of IMM and is solved on a unit circle with the Floquet exponent for the rank 1. However, this method is not easy to generalize to the case with arbitrary rank n. In this paper, we present a new systematic way to find ICB based on the loop equation.
In section 2 we present that the classical irregular conformal block F (m:n) ∆ , the inner product between irregular modules of rank m and n has the exponential form: where g s → 0 corresponds to the classical limit and f δ is finite in this limit. This is suggested in [4,5] for the regular conformal block and extended to the ICB in [6]. We demonstrate this exponential behavior for the regular and irregular case using the conformal property of the loop equation. In section 3 we present a new systematic way to analyze the classical limit of IMM and give a non-trivial example for the rank 2 partition function Z (0:2) . Further in section 4 we evaluate the partition function Z (m:n) based on the same method, which is sufficient to construct ICB. Section 5 is the conclusion and the appendix deals with the normalization of the partition function and technical details.
2 Classical form of conformal block 2.1 Setup of the formalism Regular matrix model [7,8] related with the regular conformal block is defined as the β-deformed Penner-type matrix model This potential is obtained from the correlation of K + 2 primary vertex operators (lying at 0, z 1 , · · · , z K , ∞) and screening operators (lying at z). β is related with the Virasoro screening charge b = i √ β and the Virasoro charge α of the primary operator is rescaled aŝ α = g s α. We introduce the small expansion parameter g which is related with g = ig s /2 so that √ β/g = −2b/g s . Classical limit is obtained as g s → 0 so thatα a is finite. On the other hand, b and g s are related with the Ω deformation parameter 1 = g s b and 2 = g s /b of the Nekrasov partition function [9][10][11] according to AGT conjecture [12]. Therefore, classical limit is achieved either 2 → 0 but 1 finite which is the Nekrasov-Shatashvili (NS) limit or its dual ( 1 → 0 but 2 finite). The two pictures are equivalent since Liouville theory has b → 1/b duality.
IMM is obtained in [13,14] from the colliding limit [15,16]. IMM has the same form of (2.1) but the potential is different, where m + n = K and (m + 1) primary operators are put to ∞, and (n + 1) operators to 0. The coefficientsĉ k andĉ − are given in terms of the moments;ĉ k = n r=1α r (z r ) k with k ≥ 0 andĉ − = − m a=1α a (z a ) − with > 0. The irregular partition function with the potential (2.3) will be denoted as Z (m;n) (ĉ 0 ; {ĉ k }, {ĉ − }) in the following.
The matrix model (regular or irregular) has the loop equation which presents the symmetric property where Q = b + 1/b is the background charge. W (z) and W (z, z) are the one and two point resolvents, defined as W (z) = g √ β respectively. The bracket O · · · conn denotes the connected part of the expectation value with respect to the matrix model (2.1). f (z) is the expectation value determined by the potential At the classical limit, the resolvents defined above remain finite [17]. Therefore, the loop equation (2.4) is simplified as a finite parameter at the classical/NS limit. It should be noted that (2.5) is manifestly invariant for both limits (NS and its dual). This loop equation turns into a second order differential equation (Shrödinger-like equation) if one defines Ψ(z) = exp 1 z x(z )dz : On the other hand, one may conveniently investigate the conformal block using a degenerate primary operator. In [1,18], an expectation value P (z) ≡ I (z − λ I ) is introduced in relation with the degenerate operator. It should be noted that P (z) is a polynomial of degree N , P (z) = P 0 + P 1 z + P 2 z 2 + · · · + P N −1 z N −1 + P N z N . (2.7) where N is the number of integration variables in (2.1) and P N is normalized to be 1. The wave function Ψ(z) in (2.6) is closely related with P (z). This can be seen if one notes that at the classical limit, one has [1] log Taking derivatives one has W (z) = 2 (log P (z)) = 2 P (z) P (z) and therefore, the wave-function Ψ(z) is given as This shows that the polynomial function satisfies the second order differential equation which can be check from (2.5) or equivalently from (2.6). We will use this equation to investigate the partition function and conformal block.

Classical irregular conformal block
Let us investigate the exponential behavior of the classical conformal block. Note that (1.1) is equivalent to that lim In this section we concentrate on the case of ICB. Regular conformal block is commented in sec 2.3. The explicit form of ICB is given in tems of IMM Z (m;n) [3]: , where c 0 is fixed by the neutrality condition c 0 +c ∞ +N b = Q with N the number of inserted screening operators. ICB has an extra factor e ζ (m:n) which comes from the limiting procedure z a → ∞ and z b → 0. Explicitly ζ (m:n) =ζ (m:n) /g 2 s whereζ (m:n) = min(m,n) k 2ĉ kĉ−k /k. One can confirm the exponential behavior (1.1) using the expression of ICB. We need to confirm the classical behavior It is easy to show that the first term is finite since it is given as 14) The contribution of Z (0:n) can be evaluated using f (z). Note that f (z) has a finite number of inverse powers of z: . Therefore, if one expands Eq. (2.10) in powers of z, one finds the equation has the terms running from z N −2 to z −n−1 . This provides N + n number of equations. Since there are N + n unknown variables: P 0 , P 1 , . . . , P N −1 and d 0 , d 1 , . . . , d n−1 , one can solve the equations to find d k as a function of c k 's, which are finite at the classical limit g s → 0. Once the solution of d k is found, one can find the partition function Z (0:n) using the differential equation [13] − g 2 where v k is the differential operator related with the Virasoro generator representation: Here we use the conventionĉ = 0 when c does not belong to {ĉ 0 , · · · ,ĉ n }. Once d k is known, one may rearrange (2.15) to put Since Z (0:n) only depends onĉ 0 ,ĉ 1 , . . . ,ĉ n , (2.17) is sufficient to determine Z (0:n) completely, up to the normalization factor N (0:n) which is independent ofĉ >0 : with finite H (0:n) at the classical/NS limit. In a similar way, one has f (z) = n−1 k=−m d k z −(k+2) for Z (m:n) . By identifying each coefficient of z in (2.10), there are N + m + n equations, running from z N +m−2 to z −n−1 . The number of unknown variables are also N + m + n : P 0 , P 1 , . . . , P N −1 and d −m , . . . ,d −1 , d 0 , d 1 , . . . , d n−1 . Thus solutions of d k exist as functions ofĉ k . Furthermore this coefficients allows to find Z (m:n) through the differential equation [14] −g 2 where u k is the differential operator corresponding toĉ − The solution is found similar to (2.18), with finite H (m:n) at the classical/NS limit. In addition, the conformal block (2.12) is defined as 1 ifĉ k =ĉ − = 0 for k, > 0. Therefore, the conformal block is independent of the normalization. (In appendix A we present how the normalization behaves at the classical/NS limit). Collecting all the terms, one has the classical ICB in the form of which is finite and thus, (1.1) is proved. In the following sections, we present explicit form of H (0:n) and H (m:n) which is indeed finite.

Classical regular conformal block
We may demonstrate that (1.1) holds for the classical regular conformal block too. For the regular case, we still have (2.10), but with V (z) = K a=0α a z−za and (2.23) We start with the equation of P (z) in (2.10). If one takes the residue of (2.10) around each z a , one obtains K + 1 equations for a = 0, 1, . . . , K: 2 α a P (z a ) = d a P (z a ) . (2.24) Equivalently, (2.25) Thus, one has where N β is the normalization factor independent on z a and is discussed in the appendix. One may also normalize the conformal block so that the z a -independent factor as 1. P (z) is given as the solution of (2.10). If the solution exists, the solution should be finite. There is no singularity forbidding z → z a since P (z) is the polynomial with degree One may find the explicit form of the partition function using (2.10). In this section we present how to obtain the partition function in a systematic way.
To obtain Z (0:n) , we compare each coefficient of order z l in (2.10). For the power (3.1) Here we use the notation that P a vanishes when a < 0 or a > N . The highest power z N −2 shows that which is independent ofĉ k>0 . Finding d k>0 needs algebraic manipulation. We present the case rank 2 (n = 2) explicitly, which has d 1 only. First note that the partition function is given in terms of differential equation (2.15), where h(τ ) is any function of τ ≡ĉ 2 /ĉ 2 1 which satisfies automatically v 0 (h(τ )) = 0. It is noted that the right hand side of eq. (3.5) is equivalent to H (0:2) given in (2.18) up to normalization. Eq. (3.4) requires h(τ ) to satisfy This hints thatd 1 = d 1 /ĉ 1 should be a function of τ only and one has h (τ ) =d 1 2τ 2 which can be solved as Therefore it is enough to findd 1 as a function τ . As described in appendix B we find where we neglect the τ -independent term which will be absorbed into the normalization N (0:2) . This provides the explicit partition function of rank 2: is the normalization factor independent of τ . This procedure demonstrates that finding Z (0:n) with n > 2 is straight-forward. On the other hand, it is to be noted that Z (0:n) has no filling fraction except N . This shows that Z (0:n) provides the solution of the one-cut case. In addition, it will be nice to find P (z) and d k in a more compact form.

Explicit evaluation of the partition function Z (m:n)
In this section we evaluate Z (m:n) . Its potential derivative is given as V (z) = n k=−mĉ k z k+1 and therefore, f (z) = k d k /z 2+k where k runs from −m to n − 1.
Power expansion of Eq. (2.10) provides N +m+n equations corresponding to N +m+n variables. Explicitly, for the power of z N −k−2 with −m ≤ k ≤ N +n−1 one has the algebraic equation We use the same convention in the previous section: P N = 1 and P a vanishes when a < 0 or a > N . In addition, the coefficientĉ = 0 when ≥ n + 1 or < −m.
Thus, one has the partition function (4.13) Here, N (2:2) is the normalization factor 1 . O η k are polynomials of η, and they satisfy the following conditions: 1. They can be determined completely by the group of equations (C.5) with given N , using the perturbation method; 2. They are independent of g s , i.e., they are finite because all the coefficients in the group of equations are independent of g s . In this way, (4.12) leads directly to the fact lim gs→0 g 2 s log Z (2:2) /N (2:2) → f inite.

Conclusion
Using the second order differential equation (2.10) of the polynomial P (z), we find a straightforward method to calculate classical ICB, by assuming a hierarchical ordering in c k so that P k can be treated perturbatively. Compare to known methods, this new approach is efficient since P (z) is a polynomial with a finite degree N , which leads to a finite number of equations with exact solutions. This property allows us to give a rigorous proof for the classical behavior of conformal blocks. Besides, the classical limit for Nekrasov partition function is proposed in [2] as Z N ek 2 →0 ∼ exp 1 2 W N ek , with W N ek finite. This is naturally equivalent to classical ICB's exponential behavior, through the connection of AGT conjecture. We also expect that similar discussions can be applied to W-symmetry in future.

Acknowledgements
This work is supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (NRF-2014R1A2A2A01004951).

A Normalization
In the text, we skip the normalization factor N (0:n) when the conformal block F On the other hand, the partition function (2.1) does not look invariant. However, as g s → 0 and b → ∞ (NS limit), the partition function allows the perturbation expansion, and the normalization can be taken withĉ k =0 → 0, This is given in terms of the Selberg integral [19] S N (α, δ, β) with α = 1 + √ βĉ 0 /g and δ = 1, one has 2 where A = 2ĉ 0 / . Obversely lim gs→0 g 2 s log N (0:n) is equivalent to lim β→∞ (log N (0:n) )/β. Then using the property of Gamma function log Γ(a + Z) = z log(z) − z + O(log(z)) when z is large and a is small, and making use of This means lim gs→0 g 2 s log N (0:n) (ĉ 0 ) = const. For the regular case, N β is the normalization factor independent on z a , which means it can be achieved by setting all the z a = 0. Actually According to the previous discussion we know lim gs→0 g 2 s log N β = const.

B Method to obtain d 1
The eq. (3.1) for rank 2 can be written as where we put N − k + 1 = t andã t = ( 2 (t − 2) + 2 ĉ 0 )(t − 1) − d 0 . One can simplify this if one puts r t = P t /P t−1 , where t runs from 0 to N . From the definition, one has r N +1 = 0 and r 0 = ∞. When t = 0, one finds d 1 = 2 ĉ 2 r 1 in a very compact notation. However, explicit form of r 1 as the function ofĉ 1 ,ĉ 2 is not easy to put. One way to find d 1 is to use perturbation. One may rescale r t =ĉ 1 (N + 1 − t)ξ t /(ĉ 2 t) in (B.1) to get where a t =ã t t/(N + 1 − t) and τ =ĉ 2 /ĉ 2 1 as defined in (3.5). Eq. (B.3) shows that d 1 /ĉ 1 is indeed a function of τ .
To the lowest order in τ , ξ t = 1 and d 1 /ĉ 1 = 2 N . Therefore, one may find d 1 /ĉ 1 and ξ t in powers of τd In addition, the solution ofd 1 is t-independent. Therefore, the perturbative expansion is more facilitated if the equation is set into the form where decomposition 1/ξ t = 1 + (1 − ξ t )/ξ t is used to put 1/ξ t perturbatively in t. To makẽ d 1 t-independent, one hasd (1) 1 = a N and the term in the squared bracket need to vanish at the order η , t . In this way one can findd 1 order by oder which provides the explicitd ( ) 1 with = 0, 1, 2, 3.
C Method to obtain d −1 , d 0 and d 1 For the case (m, n) = (2, 2), explicitly expanding Eq. (2.10) in powers of z, we have And in general for the power of z N −k−2 with −1 ≤ k ≤ N + 1 we have 2 (N − k + 2)ĉ 2 P N −k+2 + 2 (N − k + 1)ĉ 1 − d 1 P N −k+1 To find the d i 's, one may use perturbation in (C.5). First note that P N −1 = − I λ I while P 0 = I (−λ I ) . Therefore, P N −k can grow as k-powers of the expectation values. One may assume P N −t = O(η t ) with |η| 1. Indeed, the equation (C.5) has the solution to the lowest order, Obviously given the condition |ĉ 2 /ĉ 1 | 1, while keeping c 1 , c 0 , c −1 and c −2 in the same order, we can choose η ≡ĉ 2 /ĉ 1 , consistent with P N −t = O(η t ).